Page 86 - Probability and Statistical Inference

1. Notions of Probability 63

is a random variable with the pdf given below:

where k is a positive number and a is some number. But, it is known that P(X
≤ 2) = .9. Then,
(i) find the values of the constants k and a;
(ii) determine the expression of the df;
(iii) find the median of this distribution.
1.7.18 Suppose that the shelf life (X, measured in hours) of certain brand
of bread has the exponential pdf given by (1.7.23). We are told that 90% of
this brand of bread stays suitable for sale at the most for three days from the
moment they are put on the shelf. What percentage of this brand of bread
would last for sale at the most for two days from the moment they are put on
the shelf?
+
1.7.19 Let X be distributed as N(µ, σ ) where µ ∈ ℜ, σ ∈ ℜ .
2
(i) Suppose that P(X ≤ 10) = .25 and P(X ≤ 50) = .75. Can µ and σ
be evaluated uniquely?
(ii) Suppose that median of the distribution is 50 and P(X > 100) =
.025. Can µ and σ be evaluated uniquely?
1.7.20 Let z ∈ ℜ. Along the lines of (1.7.15)-
(1.7.16) we claim that φ(z) is a valid pdf. One can alternately use the polar
coordinates to show that by going through the following steps.
(i) Show that it is enough to prove:

(ii) Verify: Hence,

show that
(iii) In the double integral found in part (ii), use the substitutions u =
r cos(θ), υ = r sin(θ), 0 < r < ∞ and 0 < θ < 2π. Then, rewrite,
;

(iv) Evaluate explicitly to show that
(v) Does part (iii) now lead to part (ii)?
1.7.21 A soft-drink dispenser can be adjusted so that it may fill µ ounces
of the drink per cup. Suppose that the ounces of fill (X) are normally distrib-
uted with parameters µ ounce and σ = .25 ounce. Obtain the setting for µ so
that 8-ounce cups will overflow only 1.5% of the time.